Commun. Math. Phys. 101,247-282 (1985)
Communications in
Mathematical
Physics
© Springer-Verlag 1985
Renormalίzation Theory
in Four Dimensional Scalar Fields (II)
G. Gallavotti* and F. Nicolό**
Mathematics Department, Hill Center, Rutgers University, New Brunswick, NJ 08903, USA
Abstract. We interpret the results of the preceding paper (I) in terms of partial
resummations of the perturbative series for the effective interaction. As an
application we sketch how our resummation method leads to a simple
summation rule leading to a convergent expansion for the Schwinger functions
of the planar Φ^-theory.
1. Introduction
In the preceding paper [Commun. Math. Phys. 100,545-590 (1985), referred to here
as (I)] we have shown how to find "n!-bounds" on the perturbative coefficients of
the effective potential and of the Sch winger functions. The method is based on a
renormalization group approach to renormalization theory. In this paper we show
that there is a natural way of collecting together various perturbative contributions through a general analysis of summation rules for divergent series. For each
resummation we introduce some recursive relation between formal power series so
that the resummation is by definition a non-formal solution of the same recursive
relation. The similarity of this procedure with the non-perturbative methods based
on the "beta function" is manifest. As applications we discuss the theory of the
leading contribution to the effective potentials at momentum p as p-+oo and the
convergence of the planar Φ^-theory sketching a simple proof of the theorems of
'tHooft and Rivasseau [13].
Another aim of this paper is to provide some technical details not explicitly
presented in (I); they have all been collected in Appendix A. In this paper the
formulae preceded by a I [e.g. (I, 2.10)] refer to the paper (I).
Many of the ideas appearing in this paper overlap with those of references
[1-14].
* Dipartimento di matematica, II Universita di Roma, Via Raimondo, 1-00173 Roma, Italy
** Dipartimento di fisica, Universita degli studi di Roma "La Sapienza", Piazzale Aldo Moro 2,
1-00185 Roma, Italy
248
G. Gallavotti and F. Nicolό
LI. Resummatίons
The series in g, (I, 2.16), which have been implicity discussed in (I) are very likely
divergent. In this section we discuss a summation rule which transforms them into
better ones; this idea stems out of the appendix to [9] written by Rivasseau and
applied by him to study the planar Φ^-theory. This summation rule can be stated
very simply in our formalism: given a dressed tree γ we say that it is "completely
dressed" if every bifurcation of y is enclosed in a frame (otherwise we'll call it
partially dressed)
1) Completely dressed
2) partially dressed
Fig. 1.1
3) superdivergent
A dressed tree will be called "superdivergent" if it contains only dressed vertices
and its bifurcations give always rise to only two new lines. The reason of the name
"superdivergent" should be clear from Theorem 2 and the fact that these trees are
those with the greatest / for a given n (see Sect. 7 of I). We can give two essentially
equivalent, as far as the final result is concerned, summation rules which, in some
sense are at the extremes of a variety of summation rules.
Consider a dressed tree γ whose frame structure is "weakly divergent" which
means that no frame has solidly packed frames inside it, i.e. the subtree enclosed in
a given frame is never completely dressed.
Call Ψ„ the operation which associates to a dressed tree its "skeleton"
obtained by chopping off the dressed terminal lines those frames which are
enclosing completely dressed trees, together with their content of lines and internal
frames:
Renormalization Theory (ΊI)
249
itself
Fig. 1.2
Therefore Ψ^iy) is a "weakly divergent tree".
We can similarly define a "weakly 2-divergent" tree as a tree which does not
have subtrees which are superdivergent, and the operation Ψ2 over the generic
dressed tree y as the operation of chopping from γ all the superdivergent ends.
Therefore Ψ2(y) is a "weakly 2-divergent" tree. Clearly
ψ2 =ψ
Ψ22=Ψ2
(1.1)
Given now a weakly divergent tree we associate to it a tree obtained attaching to its
final lines the factor corresponding to an extra frame with a completely divergent
tree inside,
/
\
(1.2)
Therefore given a weak divergent tree (Ψ^iy) = y\ one gets from it a new tree whose
final lines have factors λf\σ\ the frequency being the frequency of the next
G. Gallavotti and F. Nicolό
250
bifurcation; this frequency will be fixed by y if the corresponding bifurcation is not
inside any other frame, otherwise it will be summed. We do that for the weak
2-divergent trees {Ψ2{y):=y% the frame we attach to any end has inside a
superdivergent tree.
The λ{kβ\σYs were called in (I) rf\σ\k) [see (I, 6.44) and the remark before
(I, 7.15)] here they are denoted by λ as they are playing the role of "bare" constants
and in the case of weak 2-divergent trees their σ correspond to superdivergent
trees, β will be as usual 0,2, 2', 4. Suppose now to drop any restriction on N and n;
therefore at the frequency k the interaction looks formally
= Σy
i
Π(γ)
Σy
k(γ) = k
Σy
V(y)\.
(1.3)
/weak 2-div^
\ trees
/
The second sum in (1.3) corresponds just to the operation of attaching at all the
ends of y all the possible superdivergent trees obtaining finally
k(y)=k
Λ /weak 2-div.\
yel
trees /
n(y)
vR(y),
(1.4)
where VR(y) differs from V(y) only because at the end of every final line off there is a
factor
(1.5)
which we describe graphically as
Fig. 1.3
Remarks, a) The combinatorial factor n(y) is the usual one [see (1,4.1)],
remembering that two final lines in y have to be considered different if their ends
have different /Γs.
b) In (1.5) Σσ r u n s o v e r the shapes of all the possible superdivergent trees;
{β
λ \h) is a formal power series in g.
From the estimates of (I) we cannot infer that (1.5) converges. Nevertheless we
iβ
can observe that λ \k) verifies a recursion relation due to the fact that a
superdivergent tree has the following structure:
Fig. 1.4
Renormalization Theory (ΊI)
251
and inside each frame this structure is repeated an arbitrary number of times. The
recursion relation is the following
(1.6)
β'.fi" 0
where g{4) = g, g{2) = m, g{2Ί = a, g{0) = 0 and we are assuming the slightly more
general interaction
Equation (1.6), written explicitly, omitting the β = 0 case which is simpler and can
be treated separately, becomes
k
ΣH
0
4V2
ΣH
0
4 2\\d(ξ-
k
0
(1.8)
4
;
^ 0
Yv-
3/ "8
2
2\ 1
It is clear that the iterative solution of (1.6) for λm(k) (β: 2,2', 4) expressed as a
power series ing(ifm = a = 0)oτg = (g, m, a) in the more general case (which can be
worked out exactly in the same way) and substituted in VR(y) [Eq. (1.4)] gives the
formal power series (1.3) for V^\φ) as before. However it might be that (1.6) admits
true non-formal solutions admitting an asymptotic expansion in g = (g, m, a), near
g=0, which necessarily must agree with the power series giving λ(k) = (λi4)(k),
{2
{2
λ \k), λ '\k)) as functions of g. Then one can state the following "summation
rule":
252
G. Gallavotti and F. Nicolό
Summation Rule. The sum
Σ y -yτ V(γ)) is by definition the expression
\Ψ2(y) = yn\y)
j
I
—— VR(y) computed with the same rule used for V(y) but with the "coupling
constants" associated to the final lines given by the true solutions λiβ\k) of
Eqs. (1.6), ...,(1.8). This will be a great improvement if the solutions have a well
defined ultraviolet limit as /c->oo.
We now investigate the resummations of the superdivergent trees in some
detail.
1.2. The Resummation of Superdivergent Trees
Let's write the asymptotic expressions of Eqs. (1.8): we define
λ{4\k) = λ(k), γ ~ 2kλ{2\k)
= μ(k), λ{2\k)
= a(k),
(1.9)
then for very large k, from (1.8) we have
+ G(4'
= G{*> 4)λ\k)
λ(k) -λ(k-l)
2)
λ(k)μ(k),
( 4 4) 2
μ(k) - μ(k -1) = - (y - 1 ) μ(k) + M ' λ (k) + M ( 2 ' 2)μ2 (k)
2
+ M ( 4 ' 2)λ(k) μ(k) + M ( 4 ' 2>)λ{k) α(fc),
( 1
'10)
where
GiiJ)=
lim G{i>j)(k),
M{iJ) = y2 lim y~2kMiiJ)(k),
k->oo
k^oo
AiiJ)=
lim
A{ίJ\k),
k-+ao
(1.11)
{Uj)
{ίJ)
{iJ
and G (h), M (h% A \h) are the coefficients of λ^ih) λ°\h) in the three
equations (1.8) respectively. The connection between Eqs. (1.10) and (1.8) together
with the explicit expressions for G{iJ\ A(iJ\ M(ίJ) is discussed in Appendix B.
We study (1.10) as if k were a continuous parameter and the finite difference
equations a system of differential equations. The possibility of doing so is also
discussed in Appendix B. Therefore we have
at
at
2
where t = k (y -1),
2
{iJ)
1
ii
G - linφ -1)" G
i)
(iJ)
and same definitions for M
and
We determine the attractive manifold in the (λ, μ, α) space. We define
2
2
μ = ^μ, α ) = i (LA + 27/lα + ^ α ) + (higher orders).
(1.13)
Renormalization Theory (II)
253
Neglecting the orders higher than 2 we have
^ = μ = Lλλ + I(h + λά) + Am
at
(1.14)
2
2
(4 4
2
(4 2
= - i (LA + 2/Λα + ,4a ) + M ' U + M ' 'Ua
+ (higher order terms),
which by consistency implies
4 = 0,
^L=M(4'4),
I = M(4>2>)
(1.15)
and
μ(Λ, α) = (M ( 4 '4 U 2 + M ( 4 ' 2 'Uα) + (higher orders).
(1.16)
Equation (1.12) becomes, neglecting terms of order higher than three
λ = G(4'4U2
ά
at
(1.17)
with (see Appendix B)
(
j(4,4)=lI(4,4)>θ5
G(4'2) = 4 I ( 2 ' ' 2 ) > 0 ,
(1.18)
and cx is a positive constant which can be explicitly computed.
Looking now for a solution (λ(/c),α(/c))->0 asfc->-αoit is easy to realize that
(1.17) can be approximated by
at
^ =-κG
at
( 44
2
U ,
κ=%,
4
(1.19)
which implies a= ~κλ and
Looking at Eq. (1.8) it follows that, with obvious definitions,
,α),
μ,α),
(1.21)
Therefore it must be possible to choose g,m,a in such a way that
{λ(k),μ(k),oί(k))->0 as fc->oo and if g is small enough we can write
g
fc'
(L22)
G. Gallavotti and F. Nicolό
254
where
(7(4,4)
4 4
(1.23)
G< >=lim
If #<0, λ(k) is defined everywhere in k, while if #>0, it has a pole for
_.
fc
ι
4 4
(G<
(G >logyV
1.3. More about Resummatίons
We want to connect now the λ{β\k) that we have discussed in the last section, with
the "form factors" rf\w, #, k) introduced in Sect. 7 of (I), [see Eq. (I, 7.18)] and
with the more standard notion of the bare coupling constant. Let us collect
together some definitions introduced in Sect. 1.1 and some new ones:
(1.2)
(1.24)
λ{nβ\k;N) =
(1.5)
Remark. The first equation is just Definition (1.2) where we have written explicitly
the dependence on the cutoff JV. The same thing can be said for the third line which
is just the definition (1.5) and the meaning of λ{nβ\k;N) is completely clear.
Nevertheless the fact that we have written the explicit JV-dependence is important
as, for instance, the general recursive relation as for instance, the analogue of (1.35)
in the non-planar case, interpreted as a true finite difference equation is different
for JVfixedfrom that obtained in the JV-» oo limit because not every tree with a high
enough number of final lines n can contribute to the kernel if JV is kept fixed. The
resummations we consider here are, anyway, those obtained in the JV->oo limit.
Comparing now the definition (1.24) with the definition of fψ\w, §\ k), we have
the following relation
rn{w) = n
#
l/(w) = / w fixed
Σf
1
Σw
cn(w) = n
ί/(w) = / wfixed
where
[see Eq. (I, 7.21)].
Σ,
k(σ) = k
v(σ) = n
f(σ) = f
yv(σ) = w
(ε(β)=ί
if
β=2
\ε(β) = O otherwise
(1.25)
Renormalization Theory (II)
255
From (1.25) and the estimates proved in Sect. 7 of (I), Theorem 2, we have
f
1
fn(w) =
=nn #
#
fn(w)
if(w) = f w fixed
0
j
Jl
i
o
(1.26)
Ji
where we used Theorem 2 and Lemmas 4 and 5 of Sect. 7 of (I). Therefore we have
the estimate
The connection between λ(β\k, N) and the "bare" coupling constant on one
side and the "running" coupling constant on the other is the following
"Bare coupling constants" = λiβ\k = N;N);
ι
{β
"Running coupling constants" = λ \k)-
)S = 0,2,2', 4,
lim λiβ)(k;N);
^ = 0,2,2^4.
The running coupling constants are the solutions of the finite difference Eq. (1.6),
or better, of the more general ones that we are going to discuss shortly. The bare
coupling constants are defined as formal power series by the following equation:
w - V{N) + Σι AN)V = - A(4)(iV, N)\dξ:
2
φψN]4:
A
<0 (1-28)
A
A
{ β)
[see Eqs. (I, 6.1) and (1,6.2), where there the coefficients λ n (N,N) were called
We are now in a better position to discuss more general resummations.
Remembering how we performed a resummation over the superdivergent trees it is
clear how one can imagine more general ones involving larger classes of
completely dressed trees. For instance a possible recursion relation is the following
one
(1.29)
which can be still generalized adding extra completely dressed trees, in the obvious
way, the next step being of course to add
1 We don't claim, the definition of "running coupling constant" used here is same as that used
by physicists, but they should be, at least, connected
256
G. Gallavotti and F. Nicolό
Σ
(1.30)
As in (1.29) the last term is of order λ3, if we can find for the equation (1.29) a
solution λ(k) = {y~ 2ε{β)kλ(β)(k)} going to zero as fc->oo, then it should coincide in
the limit fc->oo with the solution, found in the previous section, of Eq. (1.6).
We can also think of a much more drastic resummation where one resums all
the trees with frames, partially and completely dressed, to be left with only
undressed trees with R at all the bifurcations and decorated tops. A decorated top
will correspond to the complete "running coupling constant" λ{β\k), with k the
frequency of the first vertex where its line merges,
λ(β\k)=-
(1.31)
Assuming again that also in this case one could prove that γ 2ε{β)kλ{β)(k)->0 as
k-^oo one could try to proceed in this way: let V$)tR(φ[-k]) be the contribution to
V^\φ[-k]) of the undressed trees with n final lines, n S n(k), and with the coefficients
λ{β\k) instead of g{β) appended to the final lines, then [see Eqs. (1.3), (1.4)]
(1.32)
k(γ) = k
v(γ) = n
?:unframed
S we
KSO.Λ * H defined and if one expands in formal power series of g (1.32) the
standard perturbative expression is obtained again. The improvement would be of
2ε{β)k {β
course that, due to the fact y~ λ \k)-^O as fc->oo (to prove), which is what is
commonly called "asymptotic freedom," one can hope to give a well defined
meaning to the limit k-+ao of a measure
provided n(k) is chosen going to infinity as /c->oo in the appropriate way. This
program is far from being completely settled but in our opinion deserves further
investigation.
We show now that also without solving the general resummation equations
what we have proved is enough to make some definite statements about planar
theory.
1.4. Some Heuristic Results on Planar Theory
The planar Φ^-theory is defined as the perturbative Φ^-theory where only the
planar Feynman diagrams are taken into account. The obvious simplification of
Renormalization Theory (II)
257
this theory as compared with the complete one is in the fact that the number of
F-diagrams is much smaller
# [F-diagrams (top.) in Φ\ of order ή] rg cnon!,
[F-diagrams (top.) in Φ^-planar, of order ή] ^ cn0,
(1.34)
where c0 has been explicitly estimated in ('t Hooft, [13]). This suggests, remembering the nature of the problems connected to the Borel summability of the
Φ^-perturbative expansions that these results should be easier to obtain in this case
as well as stronger ones. In fact in this case one can study the most general
summation rule in the following way: Remembering the definition (1.31) of λiβ\k),
one has for it the following recursion relation:
k
oo
— + y\ y
β
0
y
2
γ
(1.35)
y
βίt...,
\y unframed
which, defining
t
(i36)
[see (I, 7.12)] can be written analytically
g^+ΣuΈn
0
Σ
2
Σ
;ku...,kJ
(1.37)
βι,...,βnku...,kn^n
where
(1.38)
v(y) = n
k(γ)=-l
γ unframed
Here y is, as in the proof of Theorem 2 of (I), a tree without frames with an R to all
its bifurcations except its lowest one, whose frequency we call h, and with root
k(y) = — 1. Σ * means that the sum is over all the γ with the constraint that the
frequencies of the bifurcations where the final lines merge and the frequency h are
kept fixed. Observe that in (1.37) the sum over h, the frequency of the
unrenormalized bifurcation, goes from 0 to k instead of from 0 to JV.
It is clear that the estimate of the kernel in (1.38) mimics exactly the estimates of
Theorems 1 and 2 of (I) with the further modification that each final line brings a
iβί
factor λ \kι) instead of g, βt being the index running over 2,2\ 4 appended to each
final line and kt the frequency of the bifurcation where the final line merges.
Therefore with the same notations of Sect. 7 of (I),
(1.39)
and 3cε > 0, such that for ε > 0
Ucnε Π
(1.40)
258
G. Gallavotti and F. Nicolό
As in the planar case Σs? nΛSc", where c is a constant proportional to c0
$ \ ^ ®
n
Π γ-MW-W»9
(1.41)
where {B}^ is the set of all the bifurcations of y. Therefore from (1.38), (1.41) and
Lemma 2 of (I)
Σ
βίt...,βnku.
Σ
7- 2 k ^ ) |^ l j ... > J/z;/c 1 ? ... ? /c / J )|^(const)",
Vfe^fc. (1.42)
.,kn
Proceeding now as in the case of the resummation of the superdivergent trees we
can write
£B Σ
Σ
y
Due to (1.42) the expression in the right-hand side of (1.43) is convergent, provided
\WXk)\£S with δ small enough.
We call ^iβ\F\');k)
the functional defined by the series in (1.43), then
F\-);k).
β
(1-44)
β)
If % (k)-+0 as k^oo (1.44) tends to coincide with Eq. (1.10) that one gets
resumming only the superdivergent trees.
Let us call J}β)(k; g) the solution of (1.44) and assume it has the same properties
that we expect for the solution of (1.10), namely that:
a) It is analytic in g9 for complex g, Re#<0 and \g\ sufficiently small.
b) It satisfies on the boundary of the analyticity region the following estimates
(1.45)
VgXφ) has to be defined as [see Eq. (1.4)]
1
ty: unf ramed
where VR(y) differs from V(y) only because at the end of every final line of y there is a
factor
(1.31)
β
Apart from that, V^\φ) has exactly the same expression as (I, 7.45) with the only
other crucial difference being that the estimates analogous to (I, 7.48), don't have
2
the factor n\ in the planar case we are considering. The generic truncated
τ
Schwinger function S (f;p) built from this Vj£\φ) is therefore analytic for g
2 This implies the convergence of the planar perturbation theory as a power series in the
JP\kys. Note that to have convergence one only needs that J}β\k, g) is small for g small, which
should be an immediate consequence of (1.44) and of the properties of the &{β\ ) functional
Renormalization Theory (II)
259
complex, R e g < 0 and \g\ sufficiently small and finally satisfies the appropriate
bounds on the boundary of this region. Reexpressing the "running" coupling
constants W\k, g) as a power series of g one gets again the formal power series of
Γ
S (/;p). Moreover the following estimate holds:
τ
k τ
s (f;P)-Σkg s Λf;pϊ
n+1 n+1
(1.47)
S(const) g (n+l)\
in the same analiticity region which follows from the assumptions a) and b) on
%β)(k;g), from (1.42) and from the estimate (1.26); (1.47) proves the Borel
summability of the renormalized planar expansion which is asymptotic to the
Schwinger function, [13].
Appendix (A)
We collect in this appendix some technical proofs of the results discussed in (I). All
the formulas quoted here refer to (I).
Lemma 1 (By Giovanni Felder, Zurich). Let # be an unlabelled Feynman graph
with n vertices, and let ybea tree with n endpoίnts. Then the number N(#, γ, {nβ}γ) of
labellings of $ compatible with y and such that for all vertices B the subgraph of $
corresponding to B has nB external lines, is bounded above by c"n(σ)expεΣnB,
for
B
all ε > 0 , and some constant cε9 if σ is the shape of the tree y.
Proof. Consider # , γ9 {nB}γ fixed. Let yB be the subtree of y with root h(B), and
NB(J) the number of ways of choosing and labelling a subgraph of § compatible
with yB and having an external line connected to the vertex J of #. Let furthermore
2?1? . . . , £ S B be the vertices following B in y. Since the subgraphs # β l , . . . , ^ β s
corresponding to Bl9 . . . , # S B have to be connected together, there exists at least
one tree diagram TB with vertices BU...,BSB
whose lines correspond to
propagators connecting @Bl,..., §BSβ. Let dBι be the number of lines of TB emerging
from Bt. We have the estimate:
Tli(dBi-l)\
(F.I)
where the last ratio is the Cayley formula for the number of rooted trees TB with
fixed coordination numbers (see Moon, J.W.: Enumerating labelled trees. In:
Graph Theory and Theoretical Physics. Harary, F. ed. London, New York:
Academic Press, 1967), and ΓL<ΛB) '
is a bound on the number of ways of
1
choosing external lines of §Bι corresponding to the lines of TB. The sum over dB. can
be performed explicitly:
NB(J) S J Γ L (max JVB|(J*))} SB(Σt nBfB ' 2,
(F.2)
260
G. Gallavotti and F. Nicolό
and using xk^
k
expε/c,
—i) = w — 1 we get:
n
(F.3)
Bey
But (Y[SBϊ\ln(σ)=
\Beγ
Jl
Π (SB\ lγ[ti B\\ (where tiB
'
j i
Beγ\
are the multiplicities of the
/
different tree shapes of the trees that start from B) is just the number of ways of
drawing the shape σ by choosing at each vertex how to order the trees starting from
it: this number is bounded by the number of ways of drawing all the trees with n
endpoints, which, by the same argument used to count the trees is bounded by cn
for some constant c. Π
Proof of Theorem 1. To prove Theorem 1 we proceed in a recursive way. Start
considering the innermost unmarked boxes, each of them has a well defined
frequency, a fixed number of vertices inside (true vertices or marked boxes shrunk
to points) and a certain number of lines going out from it. First one has to find an
estimate for the coefficient of the Wick monomial corresponding to the external
lines going out from the generic innermost unmarked box, then one has to consider
the next generation boxes (all the boxes here are unmarked). In the next step the
previous boxes behave essentially as if they were shrunk to points (due to the fact
that they are on a smaller scale as their associate frequency is higher) and therefore
as generalized vertices from which an arbitrary number lines of different type
(φ, dφ,D, S, D1) can go out. One has to prove again that the coefficient of this
generic box satisfies the estimate of the theorem, then the proof is completed as the
argument can be iterated as many times as needed until we get the final box
enclosing the whole ^.
We start considering the innermost box in Fig. la
Fig. la
The box B corresponds to the subtree of γa in the dotted frame. The contribution
from this subtree can be written in the following way
IB((P\A1X...XA1)=
ί
dη^. dηt \
n
{Λ)
dx1...dxn
(la)
k
where A has linear size y and / is the number of coordinates from which the
external legs of the box B start. PψββJφ) is defined as in Eq. (7.3): PψB% , with <gB
Renormalization Theory (II)
261
the subgraph of ^ contained in B. The estimate we need for the contribution of the
box B is an estimate for
zlix...xzlz
(Λ)n~z
(2a)
JiΛXmmmXA)(B).
B has a well defined frequency g, a fixed number of vertices inside: n, some of
them, possibly none, can be marked boxes shrunk to points, with n different
coordinates xu x 2 ,..., xn9 and each one with an index β running in the set {0,2,2'},
where here 0 means that from this vertex four lines of type φ come out, 2 means that
two lines " φ " come out and 2' that two lines of "type dφ" come out. Let's call
n0, n2, n2 the number of vertices with β: 0,2,2' respectively and lo,l2> h' the number
of half internal lines going out from the vertices of type 0,2, T respectively. The
following relations hold
4nΌ = l0 + NBo9
2n2 = l2 + NB2,
2nr = l2, + NB2,9
(3a)
where NB. are the external lines going out from the vertices of type "i". Therefore
is the total number of lines "<p" going out from B, before the
^-operation has been applied, and NB2- is the total number of lines "3φ" going out
also before the ^-operation.
Remembering the definition of the truncated expectation [see (3.8)] and that
$[β is a truncated expectation with respect to φ[ίZ], it follows that its contribution
associated to (&,B) is of the following type:
NBO,2 = NBO + NB2
C(», ZnB) x ΓΣ Π Cir i) Π (Cψ - Cri})],
(4a)
where τ represent a subset of the set of lines of Bn^ which has the property that
1J λ is a connected set (see [15]). C{n,^r\B) is a function bounded by (const)",
which takes care of the possible permutations of the half lines. The nature of the
covariances Cψ~1} or {Cψ — C(/~1}) depends on the nature of the half lines which
join to form λ or λ\ Equation (4a) and the fact that (J λ is a connected set implies
λeτ
that from the truncated expectation, associated to B, we can extract a factor
g-KγHixi, -χί2\ + \χh -χi3\ +... + !**„_! -χtn\)
?
K; > 0 ,
(5a)
where (i l 5 ..., zn) is a permutation of (1,... 5 n). This factor can always be bounded
above by
where d(xl9..., xn) is the length of the shortest polygonal path connecting all the
points x l 5 ,..,x π .
In the estimate of each covariance we have that:
1/2 line of type φ brings a factor yq,
1/2 line of type dφ brings a factor y2q.
262
G. Gallavotti and F. Nicolό
Moreover, remembering (7.12), any vertex of type 2 has an extra factor y2q. From
all that it follows that the product of covariances associated to B brings a factor we
can upper bound with
(7a)
2qn2 = cne-κγQd(xί,...,xn)yq(4n-NB-NB2>)
The ^-operation is effective when the number of external lines of B is less than or
equal to 4 (to be more precise when the "fe-dimension" of the Wick monomial is less
than or equal to 4) and in this case it substitutes some φ's or δφ's with some other
fields which have the appropriate zeroes. As the Wick monomial Pψ^lβ^iψ) has
been divided by its zeroes [see (I, 7.2)] in Fψ we have a factor of this kind
Π (yk\m - ηj\) Π (y% - ηs\) Π (/to, - nt\)2,
{l,s}
{iJ}
(8a)
{r,t}
where the first product runs over all the zeroes produced by the fields of type D
going out from B in number of nB(D), the second over all the zeroes produced by
the fields of type D1 [let their number be nB(Dι)~] and the third refers to the nB(S)
fields of type S. It follows immediately that
(9a)
and, after the ^-operation has been applied:
, = nB(φ) + nB(D) + nB(S) + nB(δφ) + nB{Dι),
2
(10a)
hold.
The factor (8a) can be written in the following way
[(8a)] =(γ-C-mnBΦ)+nBm
+
2nB(S)]). /J-J ( f | ^ . _ ^ |)Π (ffa-ηjj)
\{UΛ
ίl,s)
(f\η,-η,\)
Π
{r,t}
(lla)
and ^ l X ...x^(B) can be estimated by
,
dγ\γ . . . dηιdx1
A\ x... x A\
\Δ
\\A
. . . dxn_ι
-jyQd(Vi,...,m,x1,...,Xn-i)
\
Π (f\ηι-ηj\) Π (yqh-ηs\) Π W\ηr-rit\)2)),
U,j}
where
a) y-^y-4q(n-i)
is t h ev o
{Us}
iume factor
d u et 0
{r,t}
(12a)
))
μW| ^ ω ^ - ^ (j(β) has linear size
b) The Wick monomial has been divided to make it of "/c-dimension" = 0 by
the factor V^B+NB^^
Renormalization Theory (ΊI)
263
c) d(A1,...,Aι)is
the length of the smallest polygonal joining
The ( ) of (12a) can be estimated by
1
{Au...,Aι}.
K
J
—ΓΓ
2yqdiηi--Xn-ι)(
dη1...dηιdx1...dxn_ιe
(13a)
Therefore, using (10a),
i
Al)v
Ύ
(14a)
which is the final estimate for the innermost boxes.
We summarize here the origin of the various factors in the { } of (12a),
^Cy-4ky-4q{n~1)γk(NB+NB-\
[Volume factors and normal]
iq
k){tlB{D)+nBm
[^-operation] ~cγ~ ~
+
(15a)
2nj3(S))
.
Consider now a next-generation box B containing innermost boxes, with k its
frequency and h that of the next one surrounding B. Inside it there will be a certain
number of innermost boxes Bu B2, >. ,Br, of frequency qu ...,qr and also some
vertices: n (true vertices or marked boxes shrunk to points, as in the case of the
innermost ones)
Fig. 2a
The contribution of a innermost box JB will be written in the following way:
A
-
A
- Ίχί...dxn_ιF<§Xηί,...,ηι;xu...,xn_ι)
\A{k)(fc)|
\\A \ A ( q ) \ n - l
<k]
pl
(16a)
where
(17a)
Let us denote the contribution of the box B in the following way:
)= J dxFf{x):
(18a)
264
G. Gallavotti and F. Nicolό
where x is the set of all the coordinates inside B, including also those inside the
preceding boxes. We look for an estimate of
Λtχ...χ4β)^
ί
(19a)
dx\Fψ(x)\,
Aί X ... XAρ
where Δl9...,Δρ are tesserae of linear size y ~h, where the coordinates of the external
lines of B live. As for the innermost boxes the estimate of (19a) is made in two parts:
in the first, which is the more significant one, we have to consider all the factors y ~ka
which are produced by the covariances, the external lines, the volume factors and
the ^-operation and prove that they collect all together in the right way; the
second part consists just in proving that the remaining integration can be bounded
n+Σιnι
1
by (const) * .
a) First Part. When the Wick monomials of IBl(φ% ..., IBι(ψ) enter in the truncated
expectation at the frequency k, as '-Pψ^uβ^iψ)'' has "fc-dimension" = 0, it is clear
that the half lines coming from them in the covariance do not give any yk factor.
Vice versa the fields of the Pψtψuβ9{φ) monomials which are not contracted and
remain as external lines of B, now at the frequency h, as they have to keep their
appropriate factors in the denominator, but now with yh instead of γk, produce a
factor
-(k-h)[n B .,B{Ψ) + 2nBι,B(δφ)
+ 2nBι,B(D)
+ 3 n B [ , B ( £ *) + 3nBι,§(S)]
(20a)
where nB.B(V) is the number of external lines of type V, of Bt which are also
external lines of JJ, Vbeing φ, dφ, D,... . The vertices in B are treated exactly as for
the innermost boxes. Let ή be the number of vertices in B. Let r be the number of
the innermost boxes. Let NB be the number of external lines of B starting from the ή
vertices. Therefore the covariances of the truncated expectation S\kλ produce a
factor bounded by
Σ ιni
c
ί
+ r
'e-κy*3{x1,...tXn,ηi,...,ηr)yk(4n-NB—NB2.)
?
f>r,
where the ήu ..., ήP are coordinates of vertices in the innermost boxes
whose choice depends on (&9B,βg), d(xu ...,xή, ή1, ...,ήP) is defined
2(χ1,.
.9χn;ήi,..',ήr)=Σs\fis-'ήs+i\
+ Σj\χιJ-rij\+
ΣίK-χh+^
(21a)
Bl9...,Bn
(22a)
where the two coordinates in each | | belong always to different B/s, and the
exponential factor e~κykS{Xl""ηp) is an upper bound to the exponential decay factor
produced by the covariances associated to the contribution (g,B,βg).
<^Aίχ...χAr(B) C s e e Eq. (19a)] satisfies, before introducing the ^-operation,
άxx...άx-n
(23a)
Renormalization Theory (II)
265
The integration part of (23a) will be estimated later on.
Examine now the { } part of (21a): As
(24a)
where nB{V) are the external lines of type V, from B, starting from the vertices
1,..., n of B itself, the total number of external lines of B: Nβ satisfies
NB= Σ (ns(V) + Σt nBί,B(V)) = Σ nB(V),
(25a)
moreover, as nB(D) = ήB(Dx) = ns(S) = 0,
Σv
φ,D,D^,S}
d(V)nBuB(V)
2ns(8φ) + 2ng(D) + Zn^D1) + 3«S(S)] ,
(26a)
then
.f*%^,D>,S)w^*ι
(27a)
r
11 ι
which is also the estimate for J>Διx x Δ r (B) if (23a) is bounded by (const)
.
Before looking at the effects of the ^-operation, observe that one does not have
to worry about a possible proliferation of factors (const)ni going to the next
bifurcations of lower frequency; this is clear remembering that from each point at
most four lines can go out, which can contribute at most to four bifurcations.
We consider now what is going to change if we take into account the effect of
the ^-operation. When the Wick monomial in (18a) has "/j-dimension" ^ 4, some
lines change into other ones; this, as we are going to see, does not change the
estimate (27a) but excludes the possibility of having monomials with
"/z-dimension" ^ 4 in (18a). To prove it, examine the effect of the ^-operation only
on the { } part of (23a). Let's consider first the following example: Πβ(φ) = 2 and
nB(V) = 0VVΦφ before the ^-operation. The application of <%, see Sect. 6 of (I),
changes riβ(φ) into zero and Πβ(S) into two if the corresponding index β is 1 (see the
end of Sect. 7.4 of I); each S brings a factor y~2^k~h^ [due to its second order zero,
see (lla)] and therefore the original
y-(k-h)nB(φ)_y-(k-h)2
b
e c o m e s
y-(k-h)nB(S)y-2(k-h)nB(S)=.y-(k-h)3nB(S)
(28a)
The same happens for all the other monomials with "^-dimension ^ 4 except that
two new fields are produced: S1 and T both of "/z-dimension" = 4. Therefore after
the ^-operation is applied, again
(29a)
266
G. Gallavotti and F. Nicolό
In fact S1 appears only in the case of ΰ 1 , and as it brings an extra y~(k~h\ it follows
that after the ^-operation,
ϊntΨ1) = 3(ny ^D 1 ) + nf^S1)),
(30a)
and
y-(k-h)(3nB(Di))^y-(k-h)(3nfί(D1)+3n%(Si))y-(k-h)n§{Sί)_y-(k-hU3ns(Dί^
(31a)
Similar argument works for the T-field.
To complete the first part of the proof of Theorem 1 we have to show that the
^-operation does not change, in any case, the factor y~(/c"Λ)[ ] of Eq. (29a); the
effect of the ^-operation is, of course, that some monomials do not appear
anymore (those with "k-dimension" ^ 4), This is proved in the following way: We
indicate by n%(V) = n°(V) the external lines (with respect to B) of type V before the
^-operation is applied and n(V) those after the ^-operation which have not
changed their type; finally by nv,(V) we indicate the number of lines of type V
which before the ^-operation were of type V'(nv(V) = n(V)). Looking at the way Sk
operates [see Eqs. (6.36)-(6.40)] it is clear that the following relations hold
n\D) = n(D) + nD(dφ) + nD(S) + nD{Dx) + nD(T),
n°(S) = n(S) + ns(T);
n^S1) = n(S x ),
Let's also observe that each time a line is transformed into another one by the
^-operation, a factor y~{k~h) to some power is produced, coming from the extra
zeroes, the power being just the difference of the "fc-dimension" of V and
V'\d(V)-d(V). Therefore the factor y-<*-*H ] transforms in the following way:
-(k-h)[Σvd(V)n°(V)-4]
®\ y~(k-h)\Σvd(V)n(V) + nφ(D) + (2ndφ(Dί) + 2«D(D1)) + (2nD(S) + nφ(S)) + (3ns(T) + 2«D(Γ)) + 3nDi(S1)]
=
y-(k-h)[Σvd{V)n»(V)-4-]
We consider now how the integration in (23a) is modified. Observe that the
extra factors y~{k~h) are produced by the new zeroes which the Wick monomials
have after the ^-operation [see Eq. (lla)]. These zeroes are of the following type
Π (AηP-tfΨ1''1™
(i,r)
(i,j)
Π (yh\χa-χt\)βiatt)
s,t
sΦί
Π (yh\η\l)-χP\)yil
Up)
,
(32a)
I
(i,p)
where α, β, and γ are less than or equal to 2 and
(33a)
Renormalization Theory (II)
267
Again we rewrite these products as
[(32a)] = y ~(*-*)^«^(F)] / Π ^\η{D_η(jψ
).m \
(34a)
where the first factor in (32a) is what we have just considered in Eqs. (29a)-(31a)
and
ψ(V)^nΓ{V)-nt{V).
(35a)
n
The second factor of (34a) modifies the integration part of (23a). The theorem is
proved once the estimates of type (29a) are proved to hold for any subsequent box
B, and therefore for the whole tree, and when the remaining global integral can be
bounded by (const)".
The estimate (29a) for a generic next box B can be proved exactly as before and
produces the same estimate with the obvious changes of frequencies and indices.
We are therefore left with the estimate of an integral which, after some reflection,
one can realize is of the following type:
Π e-*^l[(zeroes)],
(36a)
where
a) λ is an internal line of ^ si is the family of all the internal lines.3
b) hλ is the frequency of the internal line λ (as previously defined). We observe
that si is completely fixed once (y, ^, βg) is assigned.
c) [(zeroes)] represent the set of zeroes which are possibly, produced at each
bifurcation by the ^-operation. We can write it
[(zeroes)] = Π ( / * l Φ ,
(37a)
I
where |Xj = <
| f — fj\ and Xis the line connecting ξ and ή which may or may not be an
internal line esi. h~λ is the frequency of the first box containing both ξ and ή.
For I(γ,&9β9) we can prove the following estimate [see (13a)]:
\I(y9&,β9)\£(consty9
(38a)
where n is the number of vertices of 0 and (const) is an appropriate constant. First of
all let's get rid of the factor [(zeroes)] in (37a). We prove the following estimate:
U(yhχ\M) Π e~ y / U | A | ^(constr
(39a)
for an appropriate constant. To prove (39a) let's observe that if λ = "(<f, JJ)" is not
an internal line, a family of internal lines exists connecting the boxes where ξ and ή
live, respectively, all with the same frequency h(B) = hχ, where B is the smallest box
containing λ and h(B) is its frequency. Graphically
3 si is a family of internal lines chosen in such a way that in each box of a well defined frequency
the internal lines of that frequency are the minimum number to make a connected path between
the inner boxes (therefore \Jtf\ = s— 1)
G. Gallavotti and F. Nicolό
268
h(B)
Fig. 3a
Let's call d(B) the polygonal drawn in Fig. 3a to connect ξ to η (remember that this
polygonal is modulo "inner boxes"); therefore the following estimate
(40a)
\Z\ίd(B)+Σid(Bi)+ ft Σjd(Bt])+ fi Σj Ίlkd(Bih)+...
holds, where d(Bι) is the length of the polygonal in the box Bt modulo the inner
boxes. Therefore the zero associated to X can be bounded by
yhl\X\ύ \yhWd(B)+
ΣiyWB)~'liKy'"d(Bi))
(41a)
Remembering that the number of zeroes for any bifurcation (box) is bounded by 3,
we can write
hl
SB
SBι
1
1
y. y.
Π {y \Ά)
λe&
'{yqiid{Biy
•π
(qii
'k y "
qiik
(42a)
qiik
\y
where the products are over the boxes hierarchically ordered from the largest one
(that corresponding to the frequency fc(y)] to the innermost ones. To bound this
factor we use the exponential part
We rearrange this expression in the following way, putting κ/2 = δ,
Be{B}
^
Renormalization Theory (II)
269
where B is the largest box, Bt are those external if we erase B and so on; qt, qtj... are
the corresponding frequencies. Now {(43a)} can be written in the following way:
{(43a)} = ( /<*> d(B) + ε Σ , y(h{B) ~qiψ d{B^) + ε Σ< Σ / 7 ( M ΰ ) " β y V y < W + ...)
V
1
Σ* ΣίJ(i-Φ
ί
1
sBi
(ΛW
1 1
"
βlj)
(ς<
+7 "
βli)
/
βiJ
))y d(By)
1 L
+ε Σky
Ί
{qij
-
qίjk
ijk
Y d(Bijk)+...
I,
(44a)
where ε is chosen such that
and
(45a)
1
1 - ε(y — 1)" * > ε <^> ε < γ " (y — 1).
This is clearly possible, therefore
ίε[yβ, J d(β i > / )+Σίy(9 ι J 9, J f c )yβ O f c d(β ι J Ϊ C )+...]]
,*r \
i '
Btj
and
[(left-hand side) (39a)] ^ ( Π xe'^
|_\j3e{β}
)3+ ( Π e~to) L
/
\B6{ΰ}
(47a)
/J
and as the number of boxes B is ^n-1, where n = φ ) [(left-hand side) (39a)]
^ (const)" as claimed.
We are left with the estimate of
io(y,9,β*)=
and
Π
\A^rιio(%),
(48a)
To do that we proceed by induction. Let's consider the lowest frequency
bifurcations (boxes) which we represent graphically in this way:
270
G. Gallavotti and F. Nicolό
Fig. 4a
Between these boxes there will be lines connecting all of them. Let's erase some of
those remaining, nevertheless, with the minimum number to form a connected
polygonal (thinking of the boxes as points). We call r\γ the endpoint of λ1>2 in the
box " ^ " all the variables in "g/' are connected to those of the other boxes only
through the line λli2, therefore the dependence on γ\ι in the integration associated
to the variables in "q^ can be eliminated by a change of variables obtaining
(49a)
where s$ is the set of all the internal lines except those we erased. sίqγ is the set of
all the internal lines associated to the subgraph @qi = @nBqi9 with Bqi the box
of frequency qv Therefore
(50a)
) ί dXqι
1
where Xqx are all the coordinates except those of the vertices in &qi.
Now we can repeat the procedure for all the remaining boxes Bqi, always
integrating first on those connected with only one other one. We end up with
(51a)
and iterating we get
Π
(52a)
3
which recalling (36a), (39a), and (48a) proves that
1(7, $, β,)^(const)"! I0(γ, <S, β9) = (const)ΐ / Π
^ (const); (const)5 = (const)".
D
\Δ(hλ)\"'
(53a)
Proof of Lemma 2. Given γ, without framed parts, with n final branches (the
number of its bifurcation is f^n— 1), let's call y the tree γ to which we have erased
Renormalization Theory (II)
271
the final lines. Then each line of y brings a factor y
frequencies of its ends. See Fig. 3a.
1/4(/ι
h
'\ where h and h! are the
Fig. 3a
Now let's move along the branches starting from the rootfc,in the direction of the
high frequencies, going back, on the opposite side, each time one arrives at the top
of a final branch off, until one is back again to k (see Fig. 3 a). The path built in that
way can be projected on the real axis: Fix on the real axis some points whose
coordinates are just the frequencies of the tree (therefore # points <;«— 1); the
path can be described as a path which goes forward and backward on the real axis
as described in Fig. 4a for the y of Fig. 3a
D
Fig. 4a
It is clear that given the path one can reconstruct f. Therefore to sum over all y is
equivalent to summing over all paths with less than n turning points whose
coordinates are fixed arbitrarily on the real axis.
The weight for each tree is
n ί
y
-ί/4- Σ i\xί-χί-ι\
i
=γ
(54a)
where L(π) is the length of the path π and x0 = k. Therefore
Σ- Σ :
xi
Xs-ι
(55a)
G. Gallavotti and F. Nicolό
272
where M and M are appropriate constants and # (n) is defined below. Let's call
φ (f) a number which tells us in how many ways we can append n final branches to
a tree y. and define
, , ,
Φ(n)=maxΦ(y).
(56a)
y
It will be proved in the next lemma that # (n) S (const)". Therefore, modulo
Lemma 2a, we have proved that
(57a)
{B}y
Lemma 2a. Given n objects and k<n boxes, if we let Jf(k).tn, denote the number of
ways one can put mi objects in the first box, m2 in the second, ...,mkin the kih one,
k
with all possible m1,...,mk,
such that Σ*m t = n, then Jf{k).n S (const)". Therefore
n-l
)S
(58a)
Σk(const)"<;(const)".
1
Proof. As
fc-l
(59a)
1
(60a)
Therefore by iteration
n
AT
^γ (k); n
= V
Λ-^mu
0
n — mjc
V
Z-ί mt - i * *
0
n— Lirni
3
V
1
Z-imo
0
(61a)
'
and defining Xj = n—
r. D
(62a)
Proof of Lemma 3. As σ(y) is fixed, to sum over γ means to sum over all the possible
frequencies. A generic σ will be of the following type:
P
Σ.s.
1
Fig. 5a
Renormalization Theory (II)
273
Therefore start summing over the frequencies of the first cluster.
1
Fig. 6a
Call IsXqi) the contribution of this cluster.
(63a)
Define θ = ^logγ9 and use the following estimate:
q+ί
i-r+l
(64a)
Applying this estimate to (63a) we get
l7
J
r
o11'"
o
lsi
ii!...iSl!
ί!
θ
(65a)
Choose now ί><θ, then
A! ;S1!
. (66a)
Now the following lemma holds:
Lemma 3a. Vs e [0, (/i + ... + fsj], f% ^ Ϊ VZ e [1, s j , ίAe following inequality holds
/l,
Jl
..,/Sl
x
γ ι
Jsi
C;
VI
-L
r
Jί'
...
Xί
~JsιJ'
•'•JSi'
\ί
^
f f'
\J\
_L
'
Jl
_i_
•••
' "
f
\\
ijsi/'
Jsim
(67a)
274
G. Gallavotti and F. Nicolό
We postpone the proof of this lemma.
Therefore
and
-by1 (ϊJjjKwj) (/1+...+/J!
Σt
Mf-
(69a)
Therefore the contribution of the cluster of marked "ends" of Fig. 6a brings the
same contribution as a simple line with a marked end,
Fig. 7a
with
/ = (/i+•••+/si)
an
d fi = y~ί/8(θ — b)~1l Y[jβ(Wj) I.
(70a)
This argument can be repeated for all other clusters and then for clusters of
clusters, getting at the very end
Λfe)
L/ifΣiT'
(71a)
which is the statement of the lemma. D
Proof of Lemma 2a. We have to prove that
This inequality has to be proven recursively in the following way:
[(left-hand side) (72a)] =
/i!-4-2
1/1 + / 2 + •••
y1
jsί-2,i
(73a)
Renormalization Theory (II)
275
Therefore once we prove that { } is less than or equal to 1, the result follows by
recursion.
We have left to prove
Λ Λ
'juht (Vl/
/ WV
?/l )/,
VΛ.ΛXI.OS/SΛ+Λ.
(74a,
Uί+J2)=j
The proof follows from the relation
Ή-.HT)-
α
Proof of Lemma 4. We have to estimate Σ ^ 1 \ remember that w fixes /, n, as the
w fixed
inner marked boxes are distributed inside the external ones and also how the
vertices are distributed inside the different boxes.
Let us assume that w is such that there are 5 external boxes and m vertices
outside the s-boxes; let's call, as usual,
f. = 1 + φ (boxes inside the i th one),
#(vertices inside the i th box),
nt=
(76a)
then
(77a)
f=Σif
1
1
Let's call Λ^/Xw) the number of subgraphs inside the zth box with the structure of
marked boxes relative to these subgraphs fixed by w. Therefore
fι
w fixed
fm
(78a)
Assume now that
J^cVirV;-/;-!)!.
(79a)
Then
9
w fixed
Z
1
t
i
1
m+s
m
fi
n
^D cro c{-Λn-f)\ύc 9{n-f)\,
(80a)
choosing c9 appropriately.
The inequality (79a) is proven by induction: Let's assume that inside the ίth box
there are st "external" boxes with / 1 ? ...,fs. boxes inside + the external one and
that nu ..., ήSι are the number of vertices inside these sf boxes respectively; finally
th
let's call mt the number of vertices inside the ί box and outside the st boxes.
Therefore
fi=l + Σjf3,
n^mt+Σjnj.
(81a)
276
G. Gallavotti and F. Nicolό
Then
ί Dm<+s'(m; + sd! Πj c%c{\- \nj-fj
-1)!
(82a)
if
10Cll
i + s(.) (m, + s; - 1 ) ) < 1,
(83a)
which is always possible by choosing appropriately c 1 0 and c n . The lemma is
proven provided
-2)!,
(84a)
which is true provided c 1 0 is chosen appropriately.
Finally we have
Lemma 5.
Σ
l^ΐ^
(7.45)
w
in, /fixed)
Proo/. Remember that fixing w means to fix
a) the total number of boxes,
b) how the boxes are included one into each other,
c) how many vertices are contained in each box.
Call symbolically {/} an arrangement of spheres (boxes) and {n} an
arrangement of vertices. Then
Σ = Σ( Σ lV
\{/} fixed
(85a)
/
We study the second sum first. If the arrangement of spheres is fixed, the regions
(differences between spheres) which belong to one sphere and not to any other
inner one are also fixed and their number is /.
Σ 1 is the number of ways one can put identical vertices inside these regions
{«}
{/} fixed
in such a way that their total number is n. This is a number bounded by the result of
Lemma 2a, therefore
Σ
{/} fixed
Λ ^ (const)".
/
(86a)
Renormalization Theory (II)
277
1
We are left with the estimate of Σ l A* arrangement of / spheres can be
{/}
described in this way: We number the spheres with an index): 0,...,/. j = 0 refers
to a fictitious sphere which encloses all the other ones. Let's order these spheres in
the following way: let's order arbitrarily the more external ones (those contained
only in the fictitious sphere s0). Let's call these ones sl9 s 2 ,..., sno, then let's consider
those inside s1 and "external" and call them sno + usno + 2, ...,s Π o + M l , then those
inside s2 and external sno+ni + u ...9sno+ni+n2
and so on, until this generation of
spheres is completed and we start considering those inside sno +1 and "external" ....
Therefore assigning the numbers {n0, nx,..., nf _ J the arrangement of spheres is
/-i
Σjnj = f
Nevertheless not any set
o
{nθ9nu...,nf-1}
describes an arrangement of spheres unless some constraints
between the n/s are satisfied. Therefore
l^(constK,
(87a)
completely
defined.
Moreover
using again Lemma 2a; Collecting (56a) and (57a) together we get the final result
Σl^cϊίΛ
88a)
choosing c 1 2 appropriately. D
Appendix B
We give an explicit expression of the coefficient G ( 4 ' 4 ) [see Eq. (1.11)] and we
examine the behaviour of G(4'4)(/c) as fc->oo.
and [see Sect. 2 of (I)]
k
0
From Eqs. (1,2.7), (1,2.8) we have
Σt$;?
o
where, looking at the equations of Sect. 2, of (I)
(3b)
278
G. Gallavotti and F. Nicolό
As for
ί&
ζ
} £ \ ( 5 b )
tζ
(4.4)= 2 W
\2
G
/Λ\2
J4
= 2
\2) ί-p-CC(z) Cl ],
where
(6b)
C(z)= lim C(t)(z)
(7b)
fc-^ oo
is defined for any z and bounded and
cx= limρ 2 C (co) (ρ)= lim(y- k z) 2 C(k)(y-fez)
ρ->0
k-*oo
pikQ
= limρ2$d*kj^->0.
(8b)
/C + 1
ρ->0
Therefore
„ ^
(44)
G
A\
2
y
Γ
dz_
J^C()
(9b)
We have also the following result: for k enough large
|G (4 4 ) (/c)-G ( 44 ) | = 0(Γ 2 ( 1 - ε ) f £ ),
ε>0.
(10b)
In fact
and
[(1 lb)] = (Cfk\z)-C{z)) (y~kz)2 Ok\y~kz) + C(z) i{y"kz)2 Ck\y-kz)-cj
k
2
m
k
. (12b)
δ
As C{z) and (y~ z) C (γ~ z) are bounded and C(z)~e~ W as |z|-*oo, <5>0,
|(llb)|^M{|C ( * ) (z)-C(z)H-e- i|ϊ| |(y-*z) 2 C ( * ) (7-*z)-c 1 |}.
Then from (4b)
\Cw(z)-C(z)\^Niy-2ke-3^.
(13b)
(14b)
Moreover
\(y-kz)2 Ok\y-kz)-Cί\S
\(y~kz)2
(15b)
Renormalization Theory (II)
279
and it is easy to realize that the first modulus can be bounded by O(y
second by ε(n), which tends to zero as n->oo, and the third one by
e'«
2/c
), the
z
,-2n
(16b)
The result (10b) is achieved observing that
J d4
ι«ι>i
SO{y-2k){z2\\og\z\\+z2\ogy2k}.
<
-2,(
-2,M-fc)_
(17b)
Proceeding in a similar way one can get explicit expressions for all the other
coefficients of (1.12) and also prove that a result analogous to (10b) holds: for k
large enough,
ε>0,
(18b)
where
), γ - 2kM(i'j\k),
AiU
(19b)
We have
G < 4
'2)=(i)(i)ίΛC(z)>
(20b)
4H
280
G. Gallavotti and F. Nicolό
We discuss now the connection between the asymptotic differential equations
(1.12) and the true ones in which we keep the dependence on k in the coefficients,
and also the connection between these results and those for the finite difference
equations. We want to prove that the solution of (1.12) gives us asymptotically the
behaviour of the solution of the true equations
at
2
^
(
4
A
2
K
+
.» ,
(1-120
This can be summarized in the following theorem, which we do not discuss here.
Theorem. Let us call n^ = (λ^, μ^, α^) a solution of (1.12) with initial data att — £0,
n 0 = woo(i0)=_g + O(^f2), where g = (g,m,a); then we have the following result: in a
ball with center n0 and radius of order g2e~ato, α>0, there exist initial data for
Eq. (1.12)' such that the corresponding solution n(t) tends to n^{t) as ί->oo, and
moreover
e-«,
(21b)
where α = 2(1— ε)logy, 0 < ε < l .
D
Therefore for t e [ί 0 , oo) we can write
e
-<".
(22b)
We finally discuss briefly the connection between Eqs. (1.10) (finite difference
equations) and (1.12). We consider the first equation of (1.10),
+ G«'2\k)λ(k)μ(k) = F(γk;λ(k),μ(k)), (23b)
λ(k)-λ(k-1) = G^\k)λ\k)
and remember that the choice of y is arbitrary provided γ>l. Therefore changing γ
in y' = y1/n, we can repeat the results and computations done before, just
substituting yf for y. Equation (23b) becomes
= λ(yk)-λ(yk-1)
λ(k)-λ(k-l)
=
λ(y'nk)-λ(y/{nk-n))
= Σi ίλ(y/{nk-{l-ί)))-λ(y/(nk-l))']
.
(24b)
1
00
k
k
Assume that λ(y ) is a C function of y . Then
) — A\y
A( y
dλ
) ^ ~7*—
ox
A
dxjx=γk-τ
ylogy=Uf)
-if)
n\dρ/ρ=k-ι/n
n
()
n\dρ
• ~'n
Jρ=k-ι/n\dxJx=ye
Renormalization Theory (II)
281
Therefore
as
n->oo, J dρ-T-.
(26b)
The right-hand side of (24b) can be worked out exactly in the same manner,
obtaining
k
f)λ
ί dρ-=
fc-i
oρ
k
J dρlogyF(ρ,λ(ρ),μ(ρ)),
(27b)
k-i
from which
— = logy F(ρ, λ(ρ), μ(ρ))
(28b)
follows, where F(fc, λ(k), μ(k)) = (y -1) F(k9 λ(k\ μ(k)) and
(29b)
Acknowledgements. We are indebted to E. Speer for discussions and detailed explanations on the
basic work of de Calan and Rivasseau, and one of us (G.G.) to V. Rivasseau for introducing him to
the theory and to the techniques of renormalization.
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Communicated by K. Osterwalder
Received May 21, 1984; in revised form January 4, 1985